Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)
A New Method Based on Undirected Graph and Polygon Triangulation to Prove the
Four-color Theorem
Xue Geng, Liu Hui, Li Shixian, Chen Wei
Division of Information Resources
China Information Technology Security Evaluation Center
Beijing, China
{xg, liuhui, lisx, chenw}@itsec.gov.cn
Abstract—The development of computer science and plane
geometry brings a new favorable opportunity for the research
about the Four-color Theorem. After converting original map
to undirected graph, this paper proposes a new method based
on undirected graph and polygon triangulation to prove the
Four-color Theorem.
Keywords- the Four-color Theorem; undirected graph;
polygon triangulation.
I.
INTRODUCTION
The Four-color Theorem, which is also known as the
Four-color Law, the Four-color Problem or the Four-color
Conjecture [1], is as famous as the Fermat Grand Theorem
and the Goldbach Conjecture. Francis Guthrie was the
exactly first people who put forward the Four-color Theorem
in 1852 when he was engaged in coloring maps [2]. However,
the formal expression about the Four-color Theorem can be
described as below:
During coloring any map, we can ensure the countries
having common boundary will be colored in different colors
just using only four kinds of colors. It is noteworthy that the
countries having only one or a finite number of common
points can not be equated to the countries having common
boundary [3].For example, country A and country C in Figure
1 can not be considered as the countries having common
boundary while country A and country B can.
So far, there are three methods to prove this theorem: the
first one is a computer assisted proof which is proposed by
Appel and Haken in 1976 [5]; the second method proposed by
Seymour and Robertson in 1994 is an improved version of
the first one and it can be checked artificially [6]; and the
latest one proposed by Gonthier in 2005 is a formal proof
with the whole logical steps [7].
However, the first one is always considered as the
official proof about the Four-color Theorem. So a lot of
researchers still want to seek a new no-computer-assisted
mathematical proof. This paper will propose a new method
utilizing polygon triangulation after converting original map
to undirected graph.
II.
CONVERSION
In order to observe the relationships of the countries
having common boundary, we convert the original map G0
(example shown as Figure 2) to undirected graph G (shown
as Figure 3) at the very beginning.
V4
V1
V5
V3
V6
V2
Figure 2. Example about the original map G0.
Figure 1. Example about countries having common boundary.
Generally speaking, we always consider the event that
famous English mathematician Cayley proposed the Fourcolor Theorem to the London Mathematical Society in 1872
as the very beginning about the experts in area of
Mathematics showing solicitude for this theorem [4]. And it
has been taken more than one century for mathematicians to
seek the way to prove it.
Figure 3. Example about undirected graph G.
Published by Atlantis Press, Paris, France.
© the authors
1966
Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)
As shown in Figure 2 and Figure 3, considering the
countries as the vertexes vi (i∈N) while considering the
common boundary as the edge ei (i ∈ N), the undirected
graph G can be described as the following expression:
G = (V, {E})
(1)
V={v1,v2,…}，E={ e 1, e 2,…}， e i=(vm, vn) (i, m,
n∈N).
Then there will be two conclusions about the undirected
graph G:
① According to the explanation about the common
boundary, one common boundary will only belong to a
couple of neighboring countries. So we can not find two
intersecting edges in the undirected graph G. And this
characteristic can be described as the following expression:
v ( ei = ( vi , v ), ej = ( v , vj ))
(2)
ei  ej (i , j ∈ N ) = 
ϕ ( any else )
② Because the neighboring relationships between the
vertexes should be the most important aspect of the
undirected graph G, we totally can make sure that there are
no straight angle in the undirected graph G. And this
characteristic can be described as the following expression:
If e i∩ e j=vk (i,j,k∈N), and e i=(vi,vk), e j=(vj,vk), then
∠vivkvj≠180°
III.
(3)
ENCLOSED UNDIRECTED GRAPH
This undirected graph G is an unclosed geometric figure.
In order to utilize the triangulation characteristic of polygon,
we must make some modifications by adding the fewest
auxiliary edges for the nearby but unconnected vertexes to
guarantee the undirected graph G become a closed geometric
figure (example shown as Figure 4).
①The Degree of every vertex in the undirected graph G’
is more than 2, and this characteristic can be described as the
following expression:
(5)
TD(vi) ≥ 2
②If we consider the vertexes of the undirected graph G’
as an intersection point of two edges, then we can consider
each mesh of the undirected graph G’ as a polygon Si (i∈N).
So the undirected graph G’ will be considered as a
combinatorial figure described as the following expression:
(6)
G’ =S1∪S2∪S3∪…
SV1∪SV2∪SV3∪…=V
(7)
SE1∪SE2∪SE3∪…=E+E’
(8)
Si=(SVi, {SEi}）.
IV.
TRIANGULATION
And next we need to make triangulation for each polygon
mesh of the undirected graph G’. However, as the rectilinear
polygon with least edges, triangle has an important inference
described as following:
Triangle is the tiniest composition units of any enclosed
rectilinear figure, in other words, a rectilinear polygon can
separate into several disjoint triangles while all the vertexes
of these triangles are the vertexes of original polygon.
It is the most common method to prove this inference
that we first consider the enclosed rectilinear figure as the
composition of a group of disjoint concave polygons and
convex polygons, and then we separate every concave
polygon into several disjoint convex polygons, and we
separate every convex polygon into several disjoint triangles
at last[8]. However, there are also other methods such as
based on determination of convex concave vertices [9], based
on BSP Tree [10] and based on Monotonic Chain [11].
According to this inference, we just can separate each
polygon mesh of the undirected graph G’ into a group of
disjoint triangles. And the vertexes of all these triangles will
be the vertexes of the undirected graph G’.
Figure 4. Example about new undirected graph G’.
And the new undirected graph G’ should also satisfy the
characteristic described as expression (2). And we can
complete this process because of the conclusion described as
expression (3).
Considering the set of adding auxiliary edges as E’, the
new undirected graph G’ can be described as the following
expression:
G’= G+E’ = (V, {E+E’})
(4)
Then there will be two conclusions about the new
undirected graph G':
Figure 5. Example about new undirected graph G”.
As shown in Figure 2, if we consider the set of adding
auxiliary edges during triangulation as E”, the new
undirected graph G” can be described as the following
expression:
G”=G’+E”= G+E’+E” = (V, {E+E’+E”})
(9)
Furthermore, if we consider each triangle mesh of the
undirected graph G” as Ti(i∈N), the new undirected graph
G” can be described as the following expression:
Published by Atlantis Press, Paris, France.
© the authors
1967
Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)
G”=T1∪T2∪T3∪…
TV1∪TV2∪TV3∪…=V
TE1∪TE2∪TE3∪…= E+E’+E”
Ti=（TVi, {TEi}）.
V.
(10)
(11)
(12)
COLORATION
After the conversion from original undirected graph G to
the new undirected graph G”, we will adopt “Triangular
Element” method, which means that we use three different
colors to color the three vertexes of a triangle mesh while the
color of third vertex is determined by the other two [12], to
color all the vertexes of undirected graph G” just beginning
with two vertexes of a random triangle mesh.
According to the expression (2), it must be noticed that
any vertex of the new undirected graph G” can be the
common vertex of several triangle meshes, but any edge of
the new undirected graph G” just can be the common edge of
two neighbor triangle meshes at most. That is to say, the
three vertexes of any triangle mesh can connect to at most
one other vertex at the same time, and they can not connect
with more than two other vertexes at the same time.
However, the only special case during the coloration will
be described as Figure 6: 2n+1 edges (n ∈ N) have one
common vertex while these 2n+1 edges are the common
edges of 2n+1 neighbor triangle meshes, then there will be
one vertex must be colored with a fourth color.
Every time the special case has come up, after using the
fourth color just once, we continue the coloration for the rest
vertexes of the new undirected graph G” by utilizing the
original three colors until we finish the whole coloration.
However, we realize the coloration for all the vertexes of
the new undirected graph G” just using no more than four
kinds of colors. According to the expression (9), there are
more edges in the new undirected graph G” than in the
original undirected graph G while both the new undirected
graph G” and the original undirected graph G have the same
number of vertexes.
So if we remove the edges of set E’ and E” in the new
undirected graph G”, we will get the original undirected
graph G, and we just reduce the relationship between the
vertexes, but there is no effect on the coloration. That means
we realize the coloration for all the countries of the original
map G0 just using no more than four colors. However, the
case show as Figure 6 proves that the least number of colors
is four.
VI.
CONCLUSION
This paper proposes a new method based on undirected
graph and polygon triangulation, and we just utilize the basic
characteristic of triangle to prove the Four-color Theorem.
Actually, with the corresponding 2D undirected graph, this
method is suitable for not only 2D but also 3D maps.
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Figure 6. Examples about the special case.
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Published by Atlantis Press, Paris, France.
© the authors
1968